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<li><a class="reference internal" href="#">Selecting the number of clusters with silhouette analysis on KMeans clustering</a></li>
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<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-cluster-plot-kmeans-silhouette-analysis-py"><span class="std std-ref">here</span></a> to download the full example code or to run this example in your browser via Binder</p>
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<div class="sphx-glr-example-title section" id="selecting-the-number-of-clusters-with-silhouette-analysis-on-kmeans-clustering">
<span id="sphx-glr-auto-examples-cluster-plot-kmeans-silhouette-analysis-py"></span><h1>Selecting the number of clusters with silhouette analysis on KMeans clustering<a class="headerlink" href="#selecting-the-number-of-clusters-with-silhouette-analysis-on-kmeans-clustering" title="Permalink to this headline">¶</a></h1>
<p>Silhouette analysis can be used to study the separation distance between the
resulting clusters. The silhouette plot displays a measure of how close each
point in one cluster is to points in the neighboring clusters and thus provides
a way to assess parameters like number of clusters visually. This measure has a
range of [-1, 1].</p>
<p>Silhouette coefficients (as these values are referred to as) near +1 indicate
that the sample is far away from the neighboring clusters. A value of 0
indicates that the sample is on or very close to the decision boundary between
two neighboring clusters and negative values indicate that those samples might
have been assigned to the wrong cluster.</p>
<p>In this example the silhouette analysis is used to choose an optimal value for
<code class="docutils literal notranslate"><span class="pre">n_clusters</span></code>. The silhouette plot shows that the <code class="docutils literal notranslate"><span class="pre">n_clusters</span></code> value of 3, 5
and 6 are a bad pick for the given data due to the presence of clusters with
below average silhouette scores and also due to wide fluctuations in the size
of the silhouette plots. Silhouette analysis is more ambivalent in deciding
between 2 and 4.</p>
<p>Also from the thickness of the silhouette plot the cluster size can be
visualized. The silhouette plot for cluster 0 when <code class="docutils literal notranslate"><span class="pre">n_clusters</span></code> is equal to
2, is bigger in size owing to the grouping of the 3 sub clusters into one big
cluster. However when the <code class="docutils literal notranslate"><span class="pre">n_clusters</span></code> is equal to 4, all the plots are more
or less of similar thickness and hence are of similar sizes as can be also
verified from the labelled scatter plot on the right.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">make_blobs</span>
<span class="kn">from</span> <span class="nn">sklearn.cluster</span> <span class="kn">import</span> <span class="n">KMeans</span>
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">silhouette_samples</span><span class="p">,</span> <span class="n">silhouette_score</span>

<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">matplotlib.cm</span> <span class="k">as</span> <span class="nn">cm</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>

<span class="nb">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</span>

<span class="c1"># Generating the sample data from make_blobs</span>
<span class="c1"># This particular setting has one distinct cluster and 3 clusters placed close</span>
<span class="c1"># together.</span>
<span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">make_blobs</span><span class="p">(</span><span class="n">n_samples</span><span class="o">=</span><span class="mi">500</span><span class="p">,</span>
                  <span class="n">n_features</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span>
                  <span class="n">centers</span><span class="o">=</span><span class="mi">4</span><span class="p">,</span>
                  <span class="n">cluster_std</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
                  <span class="n">center_box</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mf">10.0</span><span class="p">,</span> <span class="mf">10.0</span><span class="p">),</span>
                  <span class="n">shuffle</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
                  <span class="n">random_state</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>  <span class="c1"># For reproducibility</span>

<span class="n">range_n_clusters</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">]</span>

<span class="k">for</span> <span class="n">n_clusters</span> <span class="ow">in</span> <span class="n">range_n_clusters</span><span class="p">:</span>
    <span class="c1"># Create a subplot with 1 row and 2 columns</span>
    <span class="n">fig</span><span class="p">,</span> <span class="p">(</span><span class="n">ax1</span><span class="p">,</span> <span class="n">ax2</span><span class="p">)</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
    <span class="n">fig</span><span class="o">.</span><span class="n">set_size_inches</span><span class="p">(</span><span class="mi">18</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>

    <span class="c1"># The 1st subplot is the silhouette plot</span>
    <span class="c1"># The silhouette coefficient can range from -1, 1 but in this example all</span>
    <span class="c1"># lie within [-0.1, 1]</span>
    <span class="n">ax1</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="o">-</span><span class="mf">0.1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
    <span class="c1"># The (n_clusters+1)*10 is for inserting blank space between silhouette</span>
    <span class="c1"># plots of individual clusters, to demarcate them clearly.</span>
    <span class="n">ax1</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">X</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="n">n_clusters</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="mi">10</span><span class="p">])</span>

    <span class="c1"># Initialize the clusterer with n_clusters value and a random generator</span>
    <span class="c1"># seed of 10 for reproducibility.</span>
    <span class="n">clusterer</span> <span class="o">=</span> <span class="n">KMeans</span><span class="p">(</span><span class="n">n_clusters</span><span class="o">=</span><span class="n">n_clusters</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>
    <span class="n">cluster_labels</span> <span class="o">=</span> <span class="n">clusterer</span><span class="o">.</span><span class="n">fit_predict</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>

    <span class="c1"># The silhouette_score gives the average value for all the samples.</span>
    <span class="c1"># This gives a perspective into the density and separation of the formed</span>
    <span class="c1"># clusters</span>
    <span class="n">silhouette_avg</span> <span class="o">=</span> <span class="n">silhouette_score</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">cluster_labels</span><span class="p">)</span>
    <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;For n_clusters =&quot;</span><span class="p">,</span> <span class="n">n_clusters</span><span class="p">,</span>
          <span class="s2">&quot;The average silhouette_score is :&quot;</span><span class="p">,</span> <span class="n">silhouette_avg</span><span class="p">)</span>

    <span class="c1"># Compute the silhouette scores for each sample</span>
    <span class="n">sample_silhouette_values</span> <span class="o">=</span> <span class="n">silhouette_samples</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">cluster_labels</span><span class="p">)</span>

    <span class="n">y_lower</span> <span class="o">=</span> <span class="mi">10</span>
    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_clusters</span><span class="p">):</span>
        <span class="c1"># Aggregate the silhouette scores for samples belonging to</span>
        <span class="c1"># cluster i, and sort them</span>
        <span class="n">ith_cluster_silhouette_values</span> <span class="o">=</span> \
            <span class="n">sample_silhouette_values</span><span class="p">[</span><span class="n">cluster_labels</span> <span class="o">==</span> <span class="n">i</span><span class="p">]</span>

        <span class="n">ith_cluster_silhouette_values</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>

        <span class="n">size_cluster_i</span> <span class="o">=</span> <span class="n">ith_cluster_silhouette_values</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
        <span class="n">y_upper</span> <span class="o">=</span> <span class="n">y_lower</span> <span class="o">+</span> <span class="n">size_cluster_i</span>

        <span class="n">color</span> <span class="o">=</span> <span class="n">cm</span><span class="o">.</span><span class="n">nipy_spectral</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">/</span> <span class="n">n_clusters</span><span class="p">)</span>
        <span class="n">ax1</span><span class="o">.</span><span class="n">fill_betweenx</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">y_lower</span><span class="p">,</span> <span class="n">y_upper</span><span class="p">),</span>
                          <span class="mi">0</span><span class="p">,</span> <span class="n">ith_cluster_silhouette_values</span><span class="p">,</span>
                          <span class="n">facecolor</span><span class="o">=</span><span class="n">color</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="n">color</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.7</span><span class="p">)</span>

        <span class="c1"># Label the silhouette plots with their cluster numbers at the middle</span>
        <span class="n">ax1</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="o">-</span><span class="mf">0.05</span><span class="p">,</span> <span class="n">y_lower</span> <span class="o">+</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="n">size_cluster_i</span><span class="p">,</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">))</span>

        <span class="c1"># Compute the new y_lower for next plot</span>
        <span class="n">y_lower</span> <span class="o">=</span> <span class="n">y_upper</span> <span class="o">+</span> <span class="mi">10</span>  <span class="c1"># 10 for the 0 samples</span>

    <span class="n">ax1</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;The silhouette plot for the various clusters.&quot;</span><span class="p">)</span>
    <span class="n">ax1</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;The silhouette coefficient values&quot;</span><span class="p">)</span>
    <span class="n">ax1</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;Cluster label&quot;</span><span class="p">)</span>

    <span class="c1"># The vertical line for average silhouette score of all the values</span>
    <span class="n">ax1</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="n">silhouette_avg</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;red&quot;</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">)</span>

    <span class="n">ax1</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">([])</span>  <span class="c1"># Clear the yaxis labels / ticks</span>
    <span class="n">ax1</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">([</span><span class="o">-</span><span class="mf">0.1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>

    <span class="c1"># 2nd Plot showing the actual clusters formed</span>
    <span class="n">colors</span> <span class="o">=</span> <span class="n">cm</span><span class="o">.</span><span class="n">nipy_spectral</span><span class="p">(</span><span class="n">cluster_labels</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">float</span><span class="p">)</span> <span class="o">/</span> <span class="n">n_clusters</span><span class="p">)</span>
    <span class="n">ax2</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">marker</span><span class="o">=</span><span class="s1">&#39;.&#39;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">30</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.7</span><span class="p">,</span>
                <span class="n">c</span><span class="o">=</span><span class="n">colors</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;k&#39;</span><span class="p">)</span>

    <span class="c1"># Labeling the clusters</span>
    <span class="n">centers</span> <span class="o">=</span> <span class="n">clusterer</span><span class="o">.</span><span class="n">cluster_centers_</span>
    <span class="c1"># Draw white circles at cluster centers</span>
    <span class="n">ax2</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">centers</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">centers</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">marker</span><span class="o">=</span><span class="s1">&#39;o&#39;</span><span class="p">,</span>
                <span class="n">c</span><span class="o">=</span><span class="s2">&quot;white&quot;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">200</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;k&#39;</span><span class="p">)</span>

    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">centers</span><span class="p">):</span>
        <span class="n">ax2</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">c</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">c</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">marker</span><span class="o">=</span><span class="s1">&#39;$</span><span class="si">%d</span><span class="s1">$&#39;</span> <span class="o">%</span> <span class="n">i</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
                    <span class="n">s</span><span class="o">=</span><span class="mi">50</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s1">&#39;k&#39;</span><span class="p">)</span>

    <span class="n">ax2</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;The visualization of the clustered data.&quot;</span><span class="p">)</span>
    <span class="n">ax2</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;Feature space for the 1st feature&quot;</span><span class="p">)</span>
    <span class="n">ax2</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;Feature space for the 2nd feature&quot;</span><span class="p">)</span>

    <span class="n">plt</span><span class="o">.</span><span class="n">suptitle</span><span class="p">((</span><span class="s2">&quot;Silhouette analysis for KMeans clustering on sample data &quot;</span>
                  <span class="s2">&quot;with n_clusters = </span><span class="si">%d</span><span class="s2">&quot;</span> <span class="o">%</span> <span class="n">n_clusters</span><span class="p">),</span>
                 <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">,</span> <span class="n">fontweight</span><span class="o">=</span><span class="s1">&#39;bold&#39;</span><span class="p">)</span>

<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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